The Reduced and Subgame Postulate by Annie Gustafson ’24

The Reduced and Subgame Postulate by Annie Gustafson ’24, Monday May 13, 1:00 – 1:50pm, North Science Building 015, Wachenheim, Mathematics Thesis

Abstract: Suppose that, in a yes-no voting system, under some function f, f(a) < f(b) in the collection of elections in which voter c votes yes. Suppose f(a) < f(b)  in the collection of elections in which voter c votes no. If both are true, the Reduced and Subgame Postulate (RASP) asserts that, for a proper function, f(a) < f(b) the overall set of all elections. Previously, Alan Taylor proved that the Banzhaf index function satisfies RASP, while, surprisingly, the Shapley-Shubik index function fails RASP. We prove that the Deegan-Packel and Johnston index functions both fail RASP. We present weighted counterexamples to RASP for Deegan-Packel, and non-swap robust counterexamples to RASP for Shapley-Shubik and Johnston. We prove that counterexamples to RASP for the Shapley-Shubik and Johnston must fail to be swap robust and thus are not weighted.